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    Calculus Refresher
    Calculus Refresher
    Calculus Refresher
    Ebook681 pages3 hours

    Calculus Refresher

    By A. A. Klaf

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    About this ebook

    This book is unique in English as a refresher for engineers, technicians, and students who either wish to brush up their calculus or find parts of calculus unclear. It is not an ordinary textbook. It is, instead, an examination of the most important aspects of integral and differential calculus in terms of the 756 questions most likely to occur to the technical reader. It provides a very easily followed presentation and may also be used as either an introductory or supplementary textbook.
    The first part of this book covers simple differential calculus, with constants, variables, functions, increments, derivatives, differentiation, logarithms, curvature of curves, and similar topics. The second part covers fundamental ideas of integration (inspection, substitution, transformation, reduction) areas and volumes, mean value, successive and partial integration, double and triple integration. In all cases the author stresses practical aspects rather than theoretical, and builds upon such situations as might occur.
    A 50-page section illustrates the application of calculus to specific problems of civil and nautical engineering, electricity, stress and strain, elasticity, industrial engineering, and similar fields. 756 questions answered. 566 problems to measure your knowledge and improvement; answers. 36 pages of useful constants, formulae for ready reference. Index.
    LanguageEnglish
    PublisherDover Publications
    Release dateJun 8, 2012
    ISBN9780486138602
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      Book preview

      Calculus Refresher - A. A. Klaf

      SECION I SIMPLE DIFFERENTIAL CALCULUS

      CHAPTER I

      CONSTANTS—VARIABLES—FUNCTIONS-INCREMENTS

      1.  What is a constant?

      A quantity whose value is fixed.

      2.  What is a numerical or absolute constant?

      A constant that always has the same value, as 1, 2, 7, , π, ∈, etc.

      3.  What is an arbitrary constant?

      A constant that continues to have the same value throughout one problem but may have another value in a different problem. It may be represented by a letter from the beginning of the alphabet, as a, b, c, d, etc.

      4.  What is a variable?

      A quantity that may assume an indefinite number of values in the same problem and which may be represented usually by a letter from the end of the alphabet, as s, t, u, v, w, x, y, or z.

      5.  What is meant by an interval of a variable?

      The variable is considered confined to take on all values lying only between two numbers, as [a, b], a being less than b.

      6.  When is a variable said to vary continuously through

      an interval [a, b]?

      When the variable assumes in succession all possible intermediate values from a to b.

      7.  What is a function?

      A function is a relationship between variables.

      where s

      is the distance a body will fall in t sec. and g is the constant acceleration due to gravity, s is said to be a function of t.

      8.  What is a dependent variable?

      A variable whose value depends upon the value of another variable.

      If y = x² tan 30°, then y is the dependent variable. It depends on the value of the variable x.

      then s is the dependent variable. It depends on the value of the variable t

      9.  What is an independent variable?

      A variable whose value determines the value of the related dependent variable.

      If y = x² tan 30°, then x is the independent variable. It determines the value of y.

      then t is the independent variable. It determines the value of s.

      10.  What is the general relationship of the dependent and independent variables?

      The dependent variable is a function of the independent variable. In the above, y is a function of x; s is a function t.

      11.  May the dependent variable ever be taken as the independent variable?

      Frequently, when two variables are related, either may be taken as the independent variable and the other as the dependent.

      Example

      In a circle whose radius is r and whose area is A, r may be assumed as depending upon A or A as depending upon r. A change in either variable will cause a corresponding change in the other.

      12.  What is an implicit function?

      It is a function expressing an unsolved relationship between the variables.

      13.  What is an explicit function?

      A function expressing a solved relationship between the variables. One variable is solved in terms of the other.

      y = x tan 60°; y is an explicit function of x.

      x is an explicit function of y.

      x is an explicit function of y.

      z = a − y² tan x; z is an explicit function of y and x.

      The dependent variable is therefore the value of the explicit function.

      14.  What are the usual symbols for expressing a function in a general way?

      y = f(x) is read "y is a function of x" and means "y depends on the value of x."

      s = F(t) is read as is a function of t″ and means ″a depends on the value of t.″

      u = φ(ν) is read ″u is a, function of v″ and means ″u depends on the value of v.″

      Other letters may also be used to express a function. Different letters are used to represent the functions when different relations exist between the variables. The same letter is used when the same relation exists though the variables are different.

      15.  What are the usual symbols for expressing an implicit function in a general way?

      F(x, y, z);f(x, y, z); φ(x, y, z). Each expression indicates an implicit function in terms of x, y, z.

      16.  What are the usual symbols for expressing an explicit function in a general way?

      x = F(y, z); x = f(y, z); x = (y, z). Each expression denotes x as an explicit function of y and z.

      17.  How may the general symbol, as f(x), for a function be used to indicate substitutions for the variable in the function?

      If

      then

      and

      18.  When is a function said to be single valued for x − a?

      When only one value of the function corresponds to x = a. If y = Sx + 2, then y is single valued for every value of x.

      19.  When is a function multiple valued for x = a?

      When two or more values of the function correspond to x = a. If x² + y² then, for every value of x numerically less than 3, there correspond two real values of y.

      20.  What is meant by continuity and discontinuity of a function?

      The function y = 2x² is continuous for all values of x because, if x varies continuously from any value x = a to x = &p, then y will vary continuously from y = 2a² to y = 2&², and any point P(x, y) will move continuously along the graph. Here the function is defined for all values of the independent variable.

      is not defined for x is x u meaningless. There is no point on the graph for x = 0, and the function is discontinuous for x = 0. But if x increases continuously through any interval [a, b] that does not include x = 0, then y and the point P(x, y) will move continuously along the graph within that interval. For a definition, see the chapter on Limits.

      21.  What is an increment?

      An increment is any change or growth of a variable or of a function. It is the difference found by subtracting the first value from the changed, or second, value of the variable or function. The change or growth may be positive or negative according as the variable or function increases or decreases when changing.

      Example

      The area of a circle A = 7Ãr². The area A is a function of the radius r (A depends on r); any change in r is called an increment of r, and a corresponding change in A is called an increment of A.

      22.  What symbol is used to denote an increment? The Greek letter Δ (delta) denotes an increment.

      PROBLEMS

      1. How would you express the circumference of a circle as a function of its area?

      2. How would you express the area of a square as a function of its diagonal?

      3. What is the relation of the surface of a sphere to its volume?

      4. How would you express the intensity of stress on the outer fiber of a beam as a function of the bending moment and section modulus?

      5. What is the expression for the relation of the pressure head to the pressure and weight of a liquid?

      6. How would you express the horsepower of an electric power machine as a function of the electromotive force (e.m.f.) and the current?

      7. If ƒ(x) - sins, what is (a) f(0°); (b) f(30°); (c)f(sin-¹!); (d) f(cor-1 - 1); (e) ƒ(cot-1 - 1)?

      8. If F(x, y) = 5x*y + Zx²y - 6y², what is (a) F(x, -y); (b) F(-x, y); (c)F(-x,-y)1

      9. If F(x) - 2x, what is (a) F(y); (b) F(x + y)?

      10. If y = cos x, what is the explicit expression of x in terms of y?

      11. If x = 4y, what is the explicit expression of y in terms of x?

      12. If x² + y² = 8, how would you express each variable as an explicit function of the other?

      CHAPTER I

      LIMITS

      23.  What is meant by the limit of a variable?

      If a variable x approaches more and more closely a constant value c, so that c − x eventually becomes and remains less, in absolute value, than any preassigned positive number, however small, the constant c is said to be the limit of x.

      Example

      aas do all the subsequent values.

      b. Let the values of x without end. Then the limit of x is 3.

      24. How may we illustrate that a variable x is approaching 1 as a limit?

      Let line OA = 1.

      If a point x starts from O and during the first second moves half the length to x¹, during the next second half the remaining distance to x², continuing in this way to move half the remaining distance during each successive second, then the distance that the point x is from O is a variable of which OA = 1 is the limit because the difference between OA = 1 and the variable ultimately becomes and remains less than any preassigned number, however small.

      QUES. 24

      25.  Should we be concerned with the smallness of the fraction by which the variable differs from its limit?

      The smallness of the fraction by which the variable approximates the limit is of no primary importance. All we are interested in is the limit, which is fixed.

      26.  What symbols are used to indicate that a variable or a function is approaching a limit?

      x → a means "x approaches a as a limit."

      means "the limit of ƒ(x), as x approaches a as a limit, is c."

      27.  When is a variable said to increase without limit, or to become infinite?

      When the variable changes in such a manner that it becomes and remains greater than any assigned positive number, however great.

      x → ∞ means "x increases without limit, or becomes infinite."

      means "the limit of f(x), as x becomes infinite, is c."

      28.  When is a variable said to decrease without limit, or to become infinite negatively?

      When the variable changes in such a manner that it becomes and remains less than any assigned negative number, however great in absolute value.

      x → − ∞ means "x decreases without limit, or becomes infinite negatively."

      29.  What are the elementary theorems of limits?

      T1. When two variables, each approaching a limit, are equal for all their successive values, their limits are equal.

      T2. When a constant is added to a variable that approaches a limit, then the limit of their sum is the sum of the constant and the limit of the variable.

      T3. When a variable that approaches a limit is multiplied by a constant, then the limit of their product is the product of the constant and the limit of the variable.

      T4. The limit of a sum of a number of variables, each of which is approaching a limit, is the sum of their respective limits.

      T5. The limit of a product of a number of variables, each of which is approaching a limit, is the product of their respective limits.

      T6. The limit of the quotient of two variables, each of which is approaching a limit, is the quotient of their limits, except when the limit of the divisor is zero. If the limit of the divisor is zero, the limit of the quotient may have a definite finite value or the quotient may become infinite, but it is not determined by finding the quotient of the limits of the two variables.

      30.  If and x is a variable approaching 2 as a limit, what is the limit of y?

      By T2,

      and y is a variable approaching 2 as a limit.

      31.  What is the

      32.  What is the

      33.  When is a function f(x) said to be continuous and when discontinuous for x = a?

      A function ƒ(x) is continuous f or x = a when

      This means that, if the limit of the function, as x approaches a as a limit, is obtained by substituting a for x, then the function is continuous for x = a. The function is discontinuous f or x = a if this condition is not satisfied.

      Example

      It is seen that the limit of the function as x → 2 is exactly the value of the function when 2 is substituted for x. Therefore, the function x² + 3x is said to be continuous for x = 2.

      34.  Is continuous for x = 2?

      For

      Also,

      Therefore, the function is continuous for x = 2.

      35.  When a function is not already defined for x = a, is it sometimes possible to give the function such a value f or x = a as to satisfy the condition of continuity?

      Example

      is not defined f or x = 3 because there would be division by zero. But for any other value of x,

      and

      Now if we arbitrarily let the value of the function be 6 for x = 3, the function becomes continuous for this value.

      36.  When is a function continuous in an interval?

      When it is continuous for all values of x in this interval.

      37.  When is a function discontinuous for x = a?

      If f(x) becomes infinite as x approaches a as a limit, then f(x) is discontinuous for x = a, that is, when

      38. May a function have a limiting value when the independent variable becomes infinite?

      And in general, if ƒ(x) approaches the constant value c as

      Example

      As x becomes infinite in either sense, the fraction tends to zero and f(x

      39.  What are some special limits that occur frequently?

      (a

      (b

      (c

      (d

      The constant a is not zero.

      40.  What is the

      41.  What is the

      Divide the numerator and denominator by x⁴.

      Then

      As x as a limit.

      42.  What is the

      tan θ increases without limit.

      from larger values, then tan θ decreases without limit.

      has no numerical value.

      43.  What is the where x is in radians?

      For values of x approaches 1 as a limit.

      For x = 0, sin x which is indeterminate.

      PROBLEMS

      Prove that the limits of the following are as given:

      10

      CHAPTER III

      DERIVATIVES

      44.  What is the primary concern of differential calculus?

      The primary concern is the determination of the rates of change or growth of related variables, i.e., the amount of change or growth in the function (dependent variable) per unit change or growth in the value of the independent variable. Growth or change may be positive or negative.

      45.  How are independent and dependent variables represented in plotting a function?

      Abscissas represent the changing values of the independent variable, and ordinates represent the corresponding changed values of the function (dependent variable).

      46.  What is meant by the average rate of change or growth of a function in any interval?

      It is the amount of growth during the interval divided by the number of units in the interval, or the amount of growth in the value of the function (ordinate) during the interval divided by the amount of growth in the value of the independent variable during the interval.

      QUES. 46

      In the figure the increment or growth in the function (ordinate) is Δy = 5. The growth of the independent variable (abscissa) is from 10 to 20 or Δx = 10.

      In general, if y = f(x) and we give x the increment or growth Δx, then y becomes

      Subtracting

      we get

      Now divide by Δx to sret the average rate of growth.

      the average rate of growth of y with respect to x in the interval from x to x + Δx.

      47.  If a ball is thrown into the air and h = 100t − 16t² expresses the relation between

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